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APMA 3080 - Final Exam Review Questions
and Answers (100% Correct Answers) Already
Graded A+
Span Ans: The set of all linear combinations x₁u₁ + ... + xₙuₙ, where x₁, ...,
xₙ can be any real numbers.
Linear Independence Ans: The only solution to the vector equation x₁u₁
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+ ... + xₙuₙ = 0 is the trivial solution.
Linearly Dependent Ans: If a set of vectors contains the zero vector, is
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the set linearly dependent or independent?
Linearly Dependent Ans: If an nxm set of vectors in Rⁿ exists such n < m, is
the set linearly dependent or independent?
Linearly Dependent Ans: If one of the vectors in a set of vectors is a
linear combination of one of the other vectors, is the set linearly
dependent or independent?
Ax = {0} Ans: General Form of a Homogeneous Linear System
1. Closed under addition, 2. Closed under scalar multiplication Ans:
Conditions Required to Form a Transformation
One-to-One Ans: Let T be a linear transformation defined by T(x) = Ax.
The columns of A are linearly independent.
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Onto Ans: Let T be a linear transformation defined by T(x) = Ax. The
columns of A span Rⁿ.
One-to-One Ans: Let T be a linear transformation. T(x) ={0} has only the
trivial solution x = {0}.
1. Contains the zero vector, 2. Closed under addition, 3. Closed under
scalar multiplication Ans: Conditions Required to Form a Subspace
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Yes Ans: Is a span a subspace?
Null Space Ans: The set of solutions to the homogeneous linear system
Ax = {0}, where A is an nxm matrix.
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Kernel Ans: A subspace of the domain of a linear transformation T.
Range Ans: A subspace of the codomain of a linear transformation T.
Kernel Ans: The set of all vectors x such that T(x) = {0}, where T is a linear
transformation.
1. Spans the subspace, 2. Linearly independent Ans: Conditions
Required to Form a Basis
No Ans: If the number of vectors in a set in a subspace is less than the
dimension of the subspace, does the set span the subspace?
Linearly Dependent Ans: If the number of vectors in a set in a subspace
is less than the dimension of the subspace, is the set linearly independent
or dependent?
1
APMA 3080 - Final Exam Review Questions
and Answers (100% Correct Answers) Already
Graded A+
Span Ans: The set of all linear combinations x₁u₁ + ... + xₙuₙ, where x₁, ...,
xₙ can be any real numbers.
Linear Independence Ans: The only solution to the vector equation x₁u₁
© 2026 Assignment Expert
+ ... + xₙuₙ = 0 is the trivial solution.
Linearly Dependent Ans: If a set of vectors contains the zero vector, is
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the set linearly dependent or independent?
Linearly Dependent Ans: If an nxm set of vectors in Rⁿ exists such n < m, is
the set linearly dependent or independent?
Linearly Dependent Ans: If one of the vectors in a set of vectors is a
linear combination of one of the other vectors, is the set linearly
dependent or independent?
Ax = {0} Ans: General Form of a Homogeneous Linear System
1. Closed under addition, 2. Closed under scalar multiplication Ans:
Conditions Required to Form a Transformation
One-to-One Ans: Let T be a linear transformation defined by T(x) = Ax.
The columns of A are linearly independent.
, For Expert help and assignment handling,
2
Onto Ans: Let T be a linear transformation defined by T(x) = Ax. The
columns of A span Rⁿ.
One-to-One Ans: Let T be a linear transformation. T(x) ={0} has only the
trivial solution x = {0}.
1. Contains the zero vector, 2. Closed under addition, 3. Closed under
scalar multiplication Ans: Conditions Required to Form a Subspace
© 2026 Assignment Expert
Yes Ans: Is a span a subspace?
Null Space Ans: The set of solutions to the homogeneous linear system
Ax = {0}, where A is an nxm matrix.
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Kernel Ans: A subspace of the domain of a linear transformation T.
Range Ans: A subspace of the codomain of a linear transformation T.
Kernel Ans: The set of all vectors x such that T(x) = {0}, where T is a linear
transformation.
1. Spans the subspace, 2. Linearly independent Ans: Conditions
Required to Form a Basis
No Ans: If the number of vectors in a set in a subspace is less than the
dimension of the subspace, does the set span the subspace?
Linearly Dependent Ans: If the number of vectors in a set in a subspace
is less than the dimension of the subspace, is the set linearly independent
or dependent?
